Tate's theorem for the Tate cohomology (agrees with group cohomology for $r\geq1$) of finite groups states the following and is used heavily:

Let $G$ a finite group and $M$ a $G$-module and suppose that for all subgroups $H$ of $G$, $H^1_T(H,M)=0$ and $H^2_T(H,M)$ is cyclic of order $|H|$. Then for all $r$ there is an isomorphism $H^r_T(G,\mathbb{Z})\cong H^{r+2}_T(G,M)$.

See Milne's notes on 'Class field theory' or the wikipedia entry for example. This is commonly applied in the case of $G$ being a Galois group $Gal(L/K)$ and $M=L^\times$ for example.

Here's one which is key for calculations: Let $H$ be a subgroup of $G$ and $W_G(H) = N_G(H)/H$. Then the restriction map $H^*(BG) \to H^*(BH)$ maps to the invariants $(H^*(BH))^{W_G(H)}$.

When $H$ is abelian, its cohomology is well-known (polynomial tensor exterior) and thus the cohomology of $G$ is mapping to something which can in principal be computed by invariant theory.
Follow this with Quillen's theorem that the sum of these maps over all abelian subgroups has kernel which contains only nilpotent elements, and special cases such as Milgram's theorem that this is injective for symmetric groups, and you have a powerful computational tool.

It would be helpful to know what you need to know group cohomology for.

If you have an interest in pro-p or profinite groups, there's a slew of things to add on here (notably, the interpretation of the ranks of $H^1(G,F_p)$ and $H^2(G,F_p)$ as cardinalities of minimal generator and relator sets, the value of the cup product and Massey products in determining the structure of these relations, the relation with the Schur Multiplicator given by Hopf's formula above). Similarly, if you're interested in group cohomology as Galois cohomology, there's a whole field of mathematics to add on to the list. In addition to class field theory via group cohomology a la Tate, there are a few papers (I remember a particularly good one by Cornell and Rosen) that derive a large portion of a semester of algebraic number theory starting from elementary group cohomology.